PHYSICAL REVIEW E, VOLUME 64, 041603
Electrostatic correlation force of discretely charged membranes
O. González-Amezcua,1 M. Hernández-Contreras,1 and P. Pincus2
1
Departamento de Fı́sica, Centro de Investigación y Estudios Avanzados del Instituto Politécnico Nacional,
Apartado Postal 14-740, México Distrito Federal, Mexico
2
Materials Research Laboratory, University of California, Santa Barbara, California 93106
共Received 19 April 2001; revised manuscript received 18 June 2001; published 21 September 2001兲
The total force between two like charged surfaces is investigated as a function of counterion concentration
in aqueous solution and surfaces distance of separation. A smooth and a discrete density of surface charge ␴ s
lead to differences in the force distance curve at high ␴ s , which are negligible for low surface charge. The total
force per unit area with divalent counterions is an oscillating function of ␴ s . At fixed surfaces separation and
region of attraction 共increasing ␴ s 兲, there is a variation in its strength that results from a competition between
the ideal kinetic and ion-ion correlation force components as predicted from the anisotropic hypernetted chain
approximation.
DOI: 10.1103/PhysRevE.64.041603
PACS number共s兲: 68.15.⫹e, 82.65.⫹r, 68.43.⫺h
I. INTRODUCTION
Attractive forces between like charged objects due to the
correlated fluctuations of their counterion concentration
plays an important role in the adhesion of biological cell
membranes 关1–4兴. They have been also recognized as an
important mechanism for DNA to get tightly packed in a
bacterial capsid, and in eukaryotic cells 关5兴. Experimentally
these forces have been studied systematically with osmotic
stress techniques in bulk aqueous solution of DNA 关5兴, and
with the use of high resolution x-ray scattering techniques in
an overall neutral stack of synthetic charged membranes
made of a mixture of the surfactant sodium dodecyl sulfate
and the pentanol cosurfactant, in equilibrium with their dissociated counterions 关6,7兴. Recently, it was demonstrated
that these forces arise from the correlated fluctuations of
multivalent counterions that form almost two-dimensional
clouds closely bound to their compensating planar charged
surfaces 关8兴. Also condensed monovalent counterions on
highly charged surfaces with a smooth distribution of charge
lead to this attraction, as shown through numerical calculations of integral equation theory 关9–12兴 and computer simulations 关13–15兴. Using a Gaussian fluctuation theory the
asymptotic analytic expressions for this force at all separation regimes that correspond to a few angstroms were found
in a model of a pair of neutral membranes formed by an
equimolar mixture of a simple salt, which corresponds to the
case of strongly adsorbed counterions on discretely charged
surfaces 关16,17兴. It has been also used for counterions delocalized from the surfaces through an approximate mean field
distribution 关18兴. In many real situations counterions are delocalized from the surfaces and distributed quite inhomogeneously 关19兴; therefore, the correlated pressure is very sensitive to the actual ionic profile distribution and, it is expected
the pressure curve to show a more complex behavior as a
function of separation, presenting, however, the asymptotic
force laws cited above. We performed numerical calculations
of the hypernetted chain theory 共HNC兲 on the restricted
primitive model 共RPM兲 for continuous and discretely
charged membranes with their delocalized monovalent and
divalent counterions in solution, in order to determine the
total pressure between pairs of membranes under different
1063-651X/2001/64共4兲/041603共5兲/$20.00
thermodynamic conditions of charge and separation. Our calculations show that independent of the way the surface
charges are distributed, the pressure curves with a low to
moderate surface charge ␴ s coincide quantitatively, but there
appears quantitative differences at very high charge density.
The total pressure of monovalent counterions is repulsive in
the range of charge considered. For divalent ions it is found
to be repulsive for low ␴ s , and attractive for high values of
surface charge with its strength varying nonmonotically as a
function of the surface charge. These oscillations in the pressure curve derive from the competition between the ideal
kinetic and ion-ion correlations pressure contributions.
Counterion profiles at low ␴ s are found to be different for
discrete, and smooth ␴ s . However, these differences dissapear at moderate and high charge in which case the ions
deplete from the middle of the slit forming a cloud closer to
the surfaces. It is also noticeable that radial distribution functions of counterions that are located in the same layer, parallel, and just next to the surfaces display an accumulation of
counterion pairs driven by the positional correlations with
the oppositely charged neighbors.
II. MODEL SYSTEM
Two models of charged flat membranes are studied. The
aqueous solvent is treated as a continuous dielectric, ⑀
⫽78.5. Smooth and discrete distribution 共of charged spheres
on surfaces兲 are considered, neutralized by monovalent and
divalent counterions that have identical size diameter ␴
⫽4.25 Å. We shall not treat the case with added salt. Two
such surfaces are separated by a mean distance h, Fig. 1. In
this RPM model, N particles interact with the pairwise Coulomb potentials
再
qi
,
⑀
r
3D
v i 共 r 3D 兲 ⫽
⬁,
r 3D ⬎ ␴
共1兲
r 3D ⬍ ␴ ,
where q i ⫽eZ i is the total charge on particle i of valence Z i ,
and e is the elementary electric charge. r 3D is the center to
center distance of separation of two particles. Each of these
model membrane systems satisfy the electroneutrality condi-
64 041603-1
©2001 The American Physical Society
GONZÁLEZ-AMEZCUA, HERNÁNDEZ-CONTRERAS, AND PINCUS
PHYSICAL REVIEW E 64 041603
where r is the radial distance of separation between a pair of
ions in the same ionic layer and z i the midpoint perperdicular
coordinate of layer i. This integral equation is solved with
the HNC approximation
g i j 共 r,z 1 ,z 2 兲 ⫽exp关 h i j 共 r,z 1 ,z 2 兲 ⫺c i j 共 r,z 1 ,z 2 兲
⫺ ␤ v i 共 r 3D 兲 q j 兴 ,
FIG. 1. A pair of like charged flat membranes separated by
water ⑀ ⫽78.5 a mean distance h, having each one the surface
charge density ␴ s . Both the counterions of valence Zi and the
discrete charge on surfaces have diameter ␴ ⫽4.25 Å.
tion q 兰 dz ␳ (z)⫽⫺2 ␴ s , where ␳ (z) is the volume profile
distribution of counterions in the normal direction to the
planes, and ␴ s the density of surface charge. We study the
pressure P on the surfaces, and the equilibrium distribution
of counterions with the use of the anisotropic HNC theory
关10兴. In this theoretical scheme the counterion distribution is
determined from
␳ i共 z 1 兲 ⫽
e ␤␮ i
⌳ 3i
冉
冋
exp ⫺ ␤ q i ␺ 共 z 1 兲 ⫺⌺ j
冕
册
冕
dz 2 兩 z 1 ⫺z 2 兩 ␳ j 共 z 2 兲 q j ,
冕
drdz 3 c i ␥ 共 r,z 1 ,z 3 兲
⫻ ␳ ␥ 共 z 3 兲 h ␥ j 共 r,z 3 ,z 2 兲 ,
⫻
h/2
0
dr
共4兲
共7兲
dz 1 ␳ i 共 z 1 兲
冕
0
⫺h/2
dz 2 ␳ j 共 z 2 兲
⳵ ␤ u i j 共 r,z 1 ,z 2 兲
h i j 共 r,z 1 ,z 2 兲 ,
⳵z1
P core ⫽k B T2 ␲ ⌺ i, j
冕
h/2
0
dz 1
冕
0
⫺h/2
共8兲
dz 2 共 z 1 ⫺z 2 兲 ␳ i 共 z 1 兲
⫻ ␳ j 共 z 2 兲 g i j 兵 关 ␴ 2 ⫺ 共 z 1 ⫺z 2 兲 2 兴 1/2,z 1 ,z 2 其 ,
共9兲
where u i j ⫽q i q j / ⑀ r 3D , and ␳ (h/2) is the density at the midpoint between the walls, P kin is the ideal kinetic part, P el the
electrostatic correlation pressure, and P core the core-core
particle interaction pressure due to collisions of the ions.
共3兲
where ␺ (z 1 ) is the average electrostatic potential in the
counterion phase, and h the surface to surface distance of
separation. The total correlation function h i j ⫽g i j ⫺1 and the
direct correlation function c i j are determined from the
Ornstein-Zernike equation
h i j 共 r,z 1 ,z 2 兲 ⫽c i j 共 r,z 1 ,z 2 兲 ⫹⌺ ␥
冕
冕
P el ⫽⌺ i, j
共2兲
qi
qi
␴ s 共 h⫹ ␴ 兲 ⫺2 ␲ ⌺ j
⑀
⑀
共6兲
where
冊
where ⌳ i is the thermal wave length, ␮ i is the chemical
potential of the counterions between the surfaces that at thermal equilibrium is a constant independent of i. In the above
equation
⫻
P⫽ P kin ⫹ P el ⫹ P core ,
drdz 2 ␳ j 共 z 2 兲
1
⫹ 关 h i j 共 0,z 1 ,z 1 兲 ⫺c i j 共 0,z 1 ,z 1 兲兴 ,
2
q i ␺ 共 z 1 兲 ⫽⫺2 ␲
where r 3D ⫽ 冑r 2 ⫹(z 1 ⫺z 2 ) 2 , and ␤ ⫽1/k B T, k B the Boltzmann’s constant and T⫽298 K the temperature. In order to
solve Eq. 共4兲, the nonlinear Poisson-Boltzmann distribution
obtained from Eq. 共2兲 when the correlation functions are ignored is used as initial profile. Thereafter, new h i j and c i j are
determined with Eqs. 共4兲 and 共5兲, and used as inputs for
correcting the new profile given by Eq. 共2兲 until successive
iterations give updated correlation functions within 0.001%
diference from the previous solutions. Up to 41 layers were
used in the normal direction z to the surfaces, with 150 discrete points in the lateral direction r in a layer. Cuts of the
long range tails of all correlation functions due to the Coulomb interactions were performed as described in Ref. 关20兴.
Converging solutions were obtained after four steps of this
iteration scheme. The net interaction pressure P per unit area
between the surfaces is determined from
P kin ⫽k B T⌺ i ␳ i 共 h/2兲 ,
1
⫻ h 2i j 共 r,z 1 ,z 2 兲 ⫺c i j 共 r,z 1 ,z 2 兲 ⫺ ␤ v i 共 r,z 1 ,z 2 兲 q j
2
共5兲
III. RESULTS AND DISCUSSION
In this work we are interested in studying structural thermodynamic properties in models of pairs of charged membranes separated a mean distance h, made of a twodimensional ionic liquid of discrete spherical negative
charges of diameter ␴ . We will compare the same properties
for the cases of a smooth density of surface charge, and that
of a discrete charge distribution. A number of thermodynamic properties such as the pressure, free energy, internal
energy, and counterion distribution depend on the precise
knowledge of the radial distribution functions g(r,z 1 ,z 2 ) of
counterions in the gap between surfaces. We determined
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ELECTROSTATIC CORRELATION FORCE OF . . .
PHYSICAL REVIEW E 64 041603
FIG. 2. In 共a兲 are the pair correlation functions CDiv 共dashed兲
and DDiv 共dot-dashed兲 of divalent counterions at the ionic layer
z 1 / ␴ ⫽0.5, with a continuous and discrete density of charge ␴ s ⫽
⫺e/285.6 Å 2 , respectively, and surface to surface separation h
⫽1.75 Å. The continuous curve DDiv is the pair correlation function at midpoint between the surfaces z 20 / ␴ ⫽1.4. In 共b兲 the same
correlation functions are depicted as in 共a兲 for the charge density
␴ s ⫽⫺e/60 Å 2 and h⫽3.25 Å.
these structural properties for four systems: with a continuous 共C兲, and discrete 共D兲 density of surface charge ␴ s , with
monovalent 共Mon兲 and divalent 共Div兲 counterions, systems
that are denoted respectively as CMon, CDiv, DMon, and
DDiv. Three sets of parameter values were considered,
a low ␴ s ⫽⫺0.066 75 C/m2 ⫽⫺e/285.6 Å 2 , moderate
␴ s ⫽⫺0.267 C/m2 ⫽⫺e/60 Å 2 , and high ␴ s ⫽⫺0.4 C/m2
⫽⫺e/40.05 Å 2 .
We consider first the lower charge density case. In Fig. 2
we plotted g(r/ ␴ ,z 1 / ␴ ,z 2 / ␴ ) for the closest counterion
layer to the wall, and for the ionic layer at the middle of the
slit and two values of the surface to surface distance h. In
Fig. 2共a兲, curve CDiv 共dashed curve兲 is the pair correlation
function of divalent counterions in the layer at the closest
distance z 1 ⫽0.5␴ to the surfaces. This probability function
reflects the fact that for a given ␴ s continuous, an ion in layer
z 1 at the origin (r/ ␴ ⫽0) will find a second ion, in the same
layer, and located a distance r ⬘ / ␴ apart, with a probability
value that is lower than 1 共curve CDiv dashed兲. When ␴ s is
discrete, the presence of negative discrete charges on membranes increases this probability 共dot-dashed curve DDiv兲 in
the interval 0⬍r ⬘ / ␴ ⬍3. Therefore, electrostatic correlations
among positive counterions and those negative in-plane surface charges are strongest and indirectly favors counterions
to approach each other even more than when ␴ s is continuous. Discretization of in-plane charge lead to correlations of
ions that are less important for ions at the center of the slit,
z 20 / ␴ , as can be seen in Fig. 2共a兲, continuous curve DDiv,
where this correlation function shows less structure. We recall that for the case of monovalent counterions both curves
CMon and DMon do not show much differences among
them at this value of ␴ s .
An increase in the density of the surface charge to a mod-
FIG. 3. Concentration profiles ␳ (z) of counterions of valence Zi
with a continuous 共CMon, CDiv兲 and discrete 共DMon, DDiv兲 surface charge ␴ s . 共a兲 for ␴ s ⫽⫺e/285.6 Å 2 and surface- surface
seperation h⫽8.75 Å; 共b兲 for ␴ s ⫽⫺e/60 Å 2 and h⫽10.75 Å.
erate value ␴ s ⫽⫺0.267 C/m2 and small separation h
⫽3.25 Å between surfaces produces oscillations in the radial distribution functions. These are seen in Fig. 2共b兲 for
curves DDiv and CDiv of g(r/ ␴ ,z 1 / ␴ ,z 2 / ␴ ), which is the
pair correlation function of positively charged divalent counterions. Here, as in the previous case of low charge ␴ s , the
discretization of surface charge led to a distribution function
DDiv in layer z 1 / ␴ ⫽0.5 共dot-dashed curve兲 higher than
when the surface charge is continuous 共CDiv dashed兲, due to
the stronger electrostatic correlations among counterions and
surface charges. Thus, the effect of the discretization of surface charge and their correlations with ions in the bulk is to
indirectly enhance the counterions accumulation in layers
very close to the walls. Nonetheless, the radial distribution
function of ions at the midpoint of the slit, z 20 / ␴ ⫽1.4 关Fig.
2共b兲, DDiv continuous curve兴, is even higher than for ions
located at z 1 / ␴ . Thus, for surfaces separations hⰆ3.25 Å,
ions located in a layer at the midpoint in the slit display a
more structured pattern than do those ions residing in layers
closer to the surfaces at this density of charge. However, a
reverse structural effect to the one discussed just above is
found when the surfaces separation is increased to the intermediate value h⫽10.75 Å at this moderate charge ␴ s 共not
depicted兲, where in general pair correlation functions of ionic
layers closer to the walls are higher than at midway between
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GONZÁLEZ-AMEZCUA, HERNÁNDEZ-CONTRERAS, AND PINCUS
PHYSICAL REVIEW E 64 041603
FIG. 4. The total pressure between the surfaces as a function of surfaces separation h with
counterions of valence Zi; 共a兲 for surface charge
␴ s ⫽⫺e/285.6 Å 2 ; 共b兲 ␴ s ⫽⫺e/60 Å 2 , 共c兲 ␴ s
⫽⫺e/40.05 Å 2 for Zi⫽1 only; 共d兲 ␴ s ⫽
⫺e/40.05 Å 2 for Zi⫽2 only, DDiv 共CDiv兲
curve is discrete 共continuous兲 ␴ s and divalent
counterions.
walls. At the very high surface charge ␴ s ⫽⫺0.4 C/m2 , we
find a similar behavior of all correlation functions g(r,z 1 ,z 2 )
as for the case of moderate ␴ s ⫽⫺0.267 C/m2 described
before. We may summarize these qualitative observations as
follows; for very low charge ␴ s and at all separations h, the
ionic layers closer to the walls are higher 共more structured兲
than at midway distance from the surface. The same occurs
in g(r,z 1 ,z 2 ) for moderate and high surface charge if h
Ⰷ3.75 Å. These effects on the pair correlation functions are
opposite to those mentioned above when hⰆ3.75 Å for
moderate and high ␴ s , that is, g(r,z 1 ,z 2 ) is more structured
at the middle of the slit than for ions closer to the walls.
Once we determined the structural properties g(r,z 1 ,z 2 ),
we found from Eq. 共2兲 the ionic profiles ␳ i (z) in the normal
direction z to the surfaces. For all the systems the separations
were h⫽8.75 Å and 10.75 Å. These profiles ␳ i (z) are displayed in Figs. 3共a兲 for ␴ s ⫽⫺e/285.6 Å 2 , and 3共b兲 for ␴ s
⫽⫺e/60 Å 2 . From these plots we observe that discretizing
␴ s has the effect of depleting counterions from the middle of
the slit by increasing more their concentration closer to the
surfaces than in the case when ␴ s is continuous. This effect
is more important for divalent ions 关see Fig. 3共a兲 for DMon
and DDiv兴. However, such effects due to the form of ␴ s
dissapears when the surfaces get separated by few ionic diameters more 关Fig. 3共b兲兴. In general those plots show that the
profile concentration associated with monovalent ions is
higher than for divalent ones. This is so because more elementary monovalent charges are required to compensate the
surface charges. In these plots we show the modification of
the profile concentrations ␳ (z) as a function of surface separations and density of charge ␴ s . These plots show that at
larger surface separations the counterions get confined to the
surfaces, condensing closer to them, and vanish their concentration at the middle of the slit for any valence of ions. Many
quantitative differences are not observed in the profiles regardless of whether ␴ s is discrete or continuous. A similar
effect occurs if the density of surface charge is increased
while h is kept fixed. At the highest value of ␴ s considered
we noticed ␳ (z) is almost the same irrespective of the structure given to the distribution of surface charge ␴ s and valence Z i . The counterions increase their concentration profile closer to the membrane surfaces as h gets larger, and
develops oscillations for monovalent counterions.
Having addressed the problem of determining the profiles
density ␳ i (z) we calculated the total pressure from Eq. 共6兲.
At the low surface charge ␴ s ⫽⫺0.066 75 C/m2 the net
counterion pressure P for both Z i ⫽1 and 2 is repulsive at all
surface-surface separations h. It is also found that the
strength of the core-core pressure component is higher for
Z i ⫽1 than for Z i ⫽2 at all separations h and surface charge
␴ s . A fact that comes about from the larger frequency of
contacts among monovalent ions and also due to their higher
concentration. However, the electrostatic correlation pressure
component is always more important for Z i ⫽2 due to the
strongest electrostatic interactions than for Z i ⫽1. For a
given valency of ions the net pressure curves are quantitatively the same irrespective of whether ␴ s is discrete or continuous 关see Fig. 4共a兲兴 for ␴ s low.
We found above that the total pressure for low surface
charge ␴ s ⫽⫺0.066 75 C/m2 was always repulsive with any
valence of counterions. At the moderate value ␴ s ⫽
⫺0.267 C/m2 we find now that the net pressure curve of
divalent ions display already an attractive region at the very
short surfaces separation range 1 Å⬍h⬍10 Å, and turns
into repulsive for large h, vanishing for h⬎25 Å in both
cases of continuous 共effect already observed by Kjellander
et al., 关9兴兲, and discrete charge. A quite different behavior is
shown by the pressure curve of monovalent ions which at all
separations is always repulsive with a small shoulder at
about h⫽5 Å due to core-core repulsive ion interactions
关see Figs. 4共b兲 DMon and CMon兴, and lowering its strength
slowly at very large separation h. Therefore, for this surface
charge monovalent ions induce a repulsive pressure at all
separation distances and remains non-negative. Even at this
surface charge ( ␴ s ⫽⫺0.267 C/m2 ) and any valence Z i of
ions there is no quantitative differences in the net pressure
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ELECTROSTATIC CORRELATION FORCE OF . . .
PHYSICAL REVIEW E 64 041603
tions h, see Figs. 4共c兲 and 4共d兲 for monovalent and divalent
counterions, respectively. The net pressure associated to divalent ions turns always to a negative value for all h 关Fig.
4共d兲兴. Finally, Figs. 5共a兲 and 5共b兲 give us the dependence of
the net pressure P on the density of charge ␴ s for Z⫽1,2 and
h⫽5.75 Å, respectively. Here again net P of Z⫽1 and 2
shows differences at higher ␴ s . The net pressure of divalent
counterions display oscillations as a function of ␴ s , meanwhile P of monovalent ions is always an increasing function
of ␴ s and is repulsive.
IV. CONCLUSION
FIG. 5. 共a兲 The pressure curve of monovalent counterions as a
function of the density of surface charge ␴ s and fixed surfacesurface separation h⫽5.75 Å. 共b兲 The pressure of divalent counterions as a function of ␴ s for h⫽5.75 Å. The other symbols are as
in Fig. 4.
In this work we reported that discretization of the surface
charge on two equally and highly charged surfaces separated
a mean distance by a counterion solution can lead to differences on the pressure curve as a function of surfaces separation for any valence of counterions, as compared with the
pressure for surfaces with a smooth distribution of charge.
These quantitative differences disappear at low ␴ s . The
pressure curve of divalent counterions is repulsive at very
low ␴ s and turns into attractive at high charge density. In this
attractive range the strength of the total pressure varies with
increasing ␴ s due to a partial compensation of the ion-ion
correlation pressure that outweighs the ideal kinetic contribution to the total pressure, components that, however,
change their relative strengths with ␴ s . Unlike the divalent
case, the net pressure curve with monovalent counterions is
always repulsive for any charge considered.
and its components when ␴ s is continuous or discrete. The
most important effect of increasing the surface charge on the
thermodynamic properties appears in the net pressure between membranes as will be made clear below. Only at high
surface charge the effect of discretizing ␴ s becomes relevant
and leads to quantitative differences in net P at all separa-
This work was supported by a Conacyt Grant No.
32169-E, México. P.P. acknowledges support by the MRL
program of the National Science Foundation under Award
No. DMR96-32716.
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ACKNOWLEDGMENTS
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